The invention relates to the field of computer visualization of multivariate equations.
For centuries scientists and other mathematicians have described the complexity of our universe using the language of mathematics. Multivariate equations are particularly useful in describing a number of phenomena and relationships in the world, and are simply equations having a dependent variable whose outcome is functionally determined by two or more independent variables.
It is not too difficult to visualize and thus understand a multivariate equation having only two independent variables because we can plot the equation""s dependent variable on the Y-axis (vertical) and its independent variables on the X-axis (horizontal) and Z-axis (depth) to produce a 3D rendering of the equation representing all variables. It is more difficult to visualize and understand the behavior of super multivariate equations, those having three or more independent variables, because they require representations of objects having more than three dimensions. Thus, the technique described earlier has limited value for representing super multivariate equations because only two axes are available for plotting the independent variables, and the equation""s behavior relative to higher order independent variables cannot be readily displayed.
Yet there is a tremendous need in various fields to better understand the behavior of super multivariate equations. For example, in metrology the propagation of weather patterns is modeled with super multivariate equations. Many biological functions such as blood pressure are non-linearly dependent on multiple factors such as temperature, age, time of day, and chemical concentrations in the blood. All non-trivial sociological models of group behavior are dependent on more than two variables such as gross population, population density, age, and education. The behavior of most chemical reactions is dependent on more than two variables such as temperature, pressure, and chemical concentration. Many physical phenomena are non-linear and contain more than two independent variables. For example, the strength of a magnetic field at the center of a solenoid is non-linearly dependent on the number of turns in the solenoid, its diameter, and the amperage applied to the wire. From subatomic particle interactions to galactic formation, understanding the physical behavior of our universe requires an understanding of super multivariate equations.
Business relationships can also be modeled by super multivariate equations. Many econometric models are non-linear super multivariate equations dependent on variables such as inflation, GDP growth, productivity growth, population growth, consumer spending, unemployment rate, and time of year. The output of many important manufacturing and chemical refining processes are described by super multivariate equations. Efficient manufacturing and refining management requires managers to understand the relative costs, benefits, and risks of alternate production strategies. The value of most derivative financial instruments and investments is non-linearly dependent on more than two variables. The Black-Scholes option pricing equations, arbitrage free binomial and trinomial fixed income pricing models, and discounted cash flow investment models are all described by super multivariate equations.
As an example, an employee may receive stock options as part of her compensation. Typically, over time, the employee may acquire options with different strike prices and expiration dates and having a graduated vesting structure. The after tax value of this option portfolio can be modeled as a super multivariate equation with a number of independent variables including the stock price, stock price volatility, stock dividend rate, risk free interest rates, time to expiration, percentage vested, and federal and state income tax rates, which may vary relative to holding periods and income levels. The portfolio""s after-tax value can be calculated as the dependent variable of a complex non-linear super multivariate equation based on the independent variables listed above. Given the complexity, not many people can quickly understand their financial position relative to a portfolio of this nature or can best evaluate actions to maximize their after tax return. This and many other problems described by super multivariate equations could be better understood if there were better ways to represent super multivariate equations so that people could rapidly visualize the effect of changes in all of the independent variables. There is a need for more effective ways of visualizing the behavior of super multivariate equations.
The invention provides a method and system for representing multivariate equations. In one embodiment, a computer displays one or more rendered three-dimensional (3D) surfaces, which represent super multivariate equations. The surfaces are rendered on 3D Cartesian coordinates to represent the value of the equation""s dependent variable plotted over ranges for two of the equation""s independent variables (the rendered independent variables). The user can access user interface elements to modify and manipulate the values for the equation""s higher order independent variables (those which are not displayed in the Cartesian coordinates). Upon modification of these higher order independent variables, the system calculates and renders a new, corresponding 3D representation of the equation using the modified higher order independent variable values. The animated interaction of modifying the higher order independent variables by the user interface elements while observing the new representation can increase the user""s understanding of the behavior of the super multivariate equation relative to the higher order independent variable(s) being modified.
The invention takes a large step toward demystifying the behavior of super multivariate equations in a way that was not previously accomplished. The ability to understand complexity is a prerequisite for technological advancement. Tools, such as the present method and system, will enhance our understanding of complexity. The value of an intuitive tool can be significant because it permits a larger population to gain an improved understanding of the behavior of super multivariate equations in many fields.